Question

The area of a right triangle is given by the expression:

1/2 (6x² – 7x – 3).

The length of one leg is modeled with the expression:

3x + 1.

Write an expression to model
the length of the other leg.

Show all work to the best of your ability!

Answer 1

The expression that models the length of the second leg of the triangle is 2x - 3.

Recall:

The area of a triangle = $\frac{1}{2} bh$

Given:

Area of triangle = $\frac{1}{2} (6x^2 - 7x - 3)$

Length of one of the legs = 3x + 1

Therefore, it means that multiplying both legs should give us:

$6x^2 - 7x - 3$

To find the expression that models the other leg, divide $6x^2 - 7x - 3$ by 3x + 1:

• Thus:

$\frac{6x^2 - 7x - 3}{3x - 1}$

• Factorize the numerator

$\frac{6x^2 - 2x - 9x - 3}{3x - 1}\\\\\frac{2x(3x - 1) - 3(3x - 1)}{3x - 1}\\\\\frac{(2x- 3)(3x - 1)}{3x - 1}$

• Simplify

$\frac{(2x- 3)(3x - 1)}{3x - 1} = 2x- 3$

Therefore, the expression that models the length of the second leg of the triangle is 2x - 3.

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